Conjugate, or bimodal, fault patterns dominate the geological literature on shear failure. Based on Anderson’s (1905) application of the Mohr-Coulomb failure criterion, these patterns have been interpreted from all tectonic regimes, including normal, strike-slip and thrust (reverse) faulting. However, a fundamental limitation of the Mohr-Coulomb failure criterion – and others that assume faults form parallel to the intermediate principal stress, σ2 – is that only plane strain can result from slip on the conjugate faults. However, deformation in the Earth is widely accepted as being three-dimensional, with truly triaxial stresses (σ1 > σ2 > σ3) and strains. Polymodal faulting, with three or more sets of faults forming and slipping simultaneously, can generate three-dimensional strains from truly triaxial stresses. Laboratory experiments and outcrop studies have verified the occurrence of the polymodal fault patterns in nature. These fault patterns present a fundamental challenge to our understanding of shear failure in rocks (and other materials) and an opportunity to improve our understanding of seismic hazards and fluid flow in the subsurface. In this review, we assess the published evidence, theories and models for polymodal faulting before suggesting ways to produce a truly general and valid failure criterion for triaxial failure.